Category Archives: proof

Now on mathstodon.xyz

Posted on May 16, 2017 by

Christian Lawson-Perfect and Colin Wright have set up an instance of Mastodon—a decentralized, open-source Twitter clone—as a place for mathy folks to be social. It’s appropriately named mathstodon.xyz, and because it’s open-source they were able to easily hack in support. … Continue reading →

Posted in meta, proof | Tagged , , , , , , | 3 Comments

Another greedy coins game update

Posted on April 29, 2017 by

Another update on the analysis of the greedy coins game (previous posts here, here, and here). I will make another post very soon explaining how to compute optimal play. In my previous post I reported that Thibault Vroonhove had conjectured … Continue reading →

Posted in games, proof | Tagged , , , , , ,

Greedy coins game update

Posted on April 22, 2017 by

I plan to write a longer post soon, but for the moment I just wanted to provide a quick update about the greedy coins game (previous posts here and here). It turns out that the game is a lot more … Continue reading →

Posted in games, proof | Tagged , , , , , , | 18 Comments

Möbius inversion

Posted on December 14, 2016 by

In my last post we saw that , that is, the Möbius function is the inverse of with respect to Dirichlet convolution. This directly leads to an interesting principle called Möbius inversion. Möbius inversion. Suppose is defined for as the … Continue reading →

Posted in combinatorics, proof | Tagged , , , , | 3 Comments

Dirichlet convolution and the Möbius function

Posted on December 11, 2016 by

Recall from last time that the Dirichlet convolution of two functions and is written and defined by: where the sum is taken over all possible factorizations of into a product of positive integers. Last time we saw that is commutative … Continue reading →

Posted in combinatorics, proof | Tagged , , , | 1 Comment

Dirichlet convolution

Posted on December 6, 2016 by

Let and be two functions defined on the positive integers. Then the Dirichlet convolution of and , written , is another function on the positive integers, defined as follows: The sum is taken over all possible factorizations of into a … Continue reading →

Posted in combinatorics, proof | Tagged , , , | 8 Comments

The Möbius function proof, part 2 (the subset parity lemma)

Posted on December 3, 2016 by

Continuing from my previous post, we are in the middle of proving that satisfies the same equation as , that is, and that therefore for all , that is, is the sum of all the th primitive roots of unity. … Continue reading →

Posted in arithmetic, combinatorics, complex numbers, primes, proof | Tagged , , , , , , , , , | 3 Comments